### Abstract

In the paper a two-level infinitely repeated hierarchical game with one player (center) C_{0} on the first level and S_{1}...S_{n} subordinate players on the second is considered. On each stage of the game player C_{0} selects vector x=(x_{1}....x_{n}) from a given set X, in which each component represents vector of resources delivered by C_{0} to one of subordinate players, i.e. (formula presented). At the second level, S_{i} i=1,2..,n, choose the controls (formula presented), where Y_{i}(x_{i}) depends upon the choice of player C_{0}. In this game, a set of different Nash equilibrium also based on threat and punishment strategies is obtained. In one case, the center enforces special behavior of subordinate firms (vector of manufactured goods), threatening to deprive them of resources on the next steps if the subordinate firms refuse to implement the prescribed behavior. In another case, the subordinate firms can force the center to use a certain resource allocation threatening to stop production. Using different combinations of such behaviors on different stages of the game, we obtain a wide class of Nash equilibrium in the game under consideration. The cooperative version of the game is also considered. The conditions are derived under which the cooperative behavior can be supported by Nash Equilibrium or Strong Nash Equilibrium (Nash Equilibrium stable against deviations of coalitions).

Original language | English |
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Title of host publication | Mathematical Optimization Theory and Operations Research - 18th International Conference, MOTOR 2019, Proceedings |

Editors | Michael Khachay, Panos Pardalos, Yury Kochetov |

Publisher | Springer |

Pages | 685-696 |

Number of pages | 12 |

ISBN (Print) | 9783030226282 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

Event | 18th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2019 - Ekaterinburg Duration: 8 Jul 2019 → 12 Jul 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11548 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 18th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2019 |
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Country | Russian Federation |

City | Ekaterinburg |

Period | 8/07/19 → 12/07/19 |

### Fingerprint

### Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Optimization Theory and Operations Research - 18th International Conference, MOTOR 2019, Proceedings*(pp. 685-696). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11548 LNCS). Springer. https://doi.org/10.1007/978-3-030-22629-9_48

}

*Mathematical Optimization Theory and Operations Research - 18th International Conference, MOTOR 2019, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11548 LNCS, Springer, pp. 685-696, Ekaterinburg, 8/07/19. https://doi.org/10.1007/978-3-030-22629-9_48

**Equilibrium and cooperation in repeated hierarchical games.** / Petrosyan, Leon; Pankratova, Yaroslavna.

Research output › › peer-review

TY - GEN

T1 - Equilibrium and cooperation in repeated hierarchical games

AU - Petrosyan, Leon

AU - Pankratova, Yaroslavna

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In the paper a two-level infinitely repeated hierarchical game with one player (center) C0 on the first level and S1...Sn subordinate players on the second is considered. On each stage of the game player C0 selects vector x=(x1....xn) from a given set X, in which each component represents vector of resources delivered by C0 to one of subordinate players, i.e. (formula presented). At the second level, Si i=1,2..,n, choose the controls (formula presented), where Yi(xi) depends upon the choice of player C0. In this game, a set of different Nash equilibrium also based on threat and punishment strategies is obtained. In one case, the center enforces special behavior of subordinate firms (vector of manufactured goods), threatening to deprive them of resources on the next steps if the subordinate firms refuse to implement the prescribed behavior. In another case, the subordinate firms can force the center to use a certain resource allocation threatening to stop production. Using different combinations of such behaviors on different stages of the game, we obtain a wide class of Nash equilibrium in the game under consideration. The cooperative version of the game is also considered. The conditions are derived under which the cooperative behavior can be supported by Nash Equilibrium or Strong Nash Equilibrium (Nash Equilibrium stable against deviations of coalitions).

AB - In the paper a two-level infinitely repeated hierarchical game with one player (center) C0 on the first level and S1...Sn subordinate players on the second is considered. On each stage of the game player C0 selects vector x=(x1....xn) from a given set X, in which each component represents vector of resources delivered by C0 to one of subordinate players, i.e. (formula presented). At the second level, Si i=1,2..,n, choose the controls (formula presented), where Yi(xi) depends upon the choice of player C0. In this game, a set of different Nash equilibrium also based on threat and punishment strategies is obtained. In one case, the center enforces special behavior of subordinate firms (vector of manufactured goods), threatening to deprive them of resources on the next steps if the subordinate firms refuse to implement the prescribed behavior. In another case, the subordinate firms can force the center to use a certain resource allocation threatening to stop production. Using different combinations of such behaviors on different stages of the game, we obtain a wide class of Nash equilibrium in the game under consideration. The cooperative version of the game is also considered. The conditions are derived under which the cooperative behavior can be supported by Nash Equilibrium or Strong Nash Equilibrium (Nash Equilibrium stable against deviations of coalitions).

KW - Cooperation

KW - Nash equilibrium

KW - Repeated hierarchical game

UR - http://www.scopus.com/inward/record.url?scp=85067671871&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-22629-9_48

DO - 10.1007/978-3-030-22629-9_48

M3 - Conference contribution

SN - 9783030226282

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 685

EP - 696

BT - Mathematical Optimization Theory and Operations Research - 18th International Conference, MOTOR 2019, Proceedings

A2 - Khachay, Michael

A2 - Pardalos, Panos

A2 - Kochetov, Yury

PB - Springer

ER -